300 XIV INTEGRATION IN LOCALLY COMPACT GROUPS
(a) Let/be a continuous function on G, with values in [0, 1] and support contained
in V, and such that/(*?)> 0. Put
Show that for each z e G we have ||Y(Z)# — g\\ <; c\\y(z)/-/||, where c = \f(s) df$(s),
and that g(z) < g(e) if z ^ e. (If not, we should have y(z~1)f=f, which would imply
that z* e V for all k.)
(b) Let (fn) be a uniformly bounded equicontinuous family of positive functions
with support contained in V, and suppose that for each n there exists a subset An <=• V
such that ||Y(*)/« ~/J ^ M for all n and all z E An. Show that the set of functions
is equicontinuous, and that the same is true of the set of functions j(z)gn — gn where
z e An , for each n.
10. Let G be a locally compact group and let ut v be two positive bounded functions on G.
For each x e G let
w(x) = sup u(y)v(y'lx).
yeG
(a) Show that w(x) = sup u(xy~l)v(y), and that for each s e G we have
yeG
If A, B are the supports of u, v respectively, show that the support of w is contained
in the closure of AB.
(b) Suppose that v(x~ l) = v(x). Show that, for all x, x' in G, we have
MX)- w(x')\ ^||
(Observe that for each y e G we have
Y(*'MI £ u(y)(v(y~lx) -
In particular, if v is continuous and compactly supported, then so is w.
11. Let G be a locally compact group, and V a symmetric compact neighborhood of e in
G such that V2 contains no subgroup =£{e}. For each integer /, let U/ be the set of
points x such that xk e V for 1 ^ k <£ /. As / -» + °o, the sets U/ form a fundamental
system of neighborhoods of e in G (Section 12.9, Problem 6b)). Let nt be the largest
integer n such that U" <= V. Define a function wf on G as follows :
(a) Show that for all ;*: € U( we haveinto m parts by means of an increasing sequence